Here we can only speak for ourselves, but by default, when the chat service is offline then absolutely nothing related to chat (i.e. chat button etc.) is shown on the website.
The reason for this is that whenever a visitor sees a live chat button on a website he also expects to be able to use it. And the last thing retailers want to do is to disappoint their customers by promising them something and then failing to deliver.
Naturally, it is always possible to use other type of actions on the website when the chat service is offline. For example, a contact information form for collecting leads or a simple pop up informing the visitor about live chat service hours, can be placed on the website instead of chat, when there are no chat agents online. This way companies can make sure your service quality doesn’t suddenly drop after hours.
Another options companies can utilise is to hire an outsourced chat service provider to take care of incoming chat conversations when all internal chat agents are offline.